Lecture #2: Review of Spin Physics • Topics – Spin – The Nuclear Spin Hamiltonian – Coherences • References – Levitt, Spin Dynamics 1 Nuclear Spins • Protons (as well as electrons and neutrons) possess intrinsic angular momentum called “spin”, which gives rise to a magnetic dipole moment. Plank’s constant 1 µ = γ! 2 spin gyromagnetic ratio • In a magnetic field, the spin precesses around the applied field. z Precession frequency θ µ ω 0 ≡ γB0 B = B0zˆ Note: Some texts use ω0 = -gB0. y ! ! Energy = −µ ⋅ B = − µ Bcosθ x How does the concept of energy differ between classical and quantum physics • Question: What magnetic (and electric?)€ fields influence nuclear spins? 2 The Nuclear Spin Hamiltonian • Hˆ is the sum of different terms representing different physical interactions. ˆ ˆ ˆ ˆ H = H1 + H 2 + H 3 +! Examples: 1) interaction of spin with B0 2) interactions with dipole fields of other nuclei 3) nuclear-electron couplings • In general,€ we can think of an atomic€ nucleus as a lumpy magnet with a (possibly non-uniform) positive electric charge • The spin Hamiltonian contains terms which describe the orientation dependence of the nuclear energy The nuclear magnetic moment interacts with magnetic fields Hˆ = Hˆ elec + Hˆ mag The nuclear electric charge interacts with electric fields 3 € Electromagnetic Interactions • Magnetic interactions magnetic moment ! ! ! ! Hˆ mag ˆ B Iˆ B = −µ ⋅ = −γ" ⋅ local magnetic field quadrapole • Electric interactions monopole dipole Nuclear€ electric charge distributions can be expressed as a sum of multipole components. ! (0) ! (1) ! (2) ! C (r ) = C (r ) + C (r ) + C (r ) +" Symmetry properties: C(n)=0 for n>2I and odd interaction terms disappear Hence, for spin-½ nuclei there are no electrical ˆ elec H = 0 (for spin I€= 1/2) energy terms that depend on orientation or internal nuclear structure, and they behaves exactly like ˆ elec H ≠ 0 (for spin I >1/2) point charges! Nuclei with spin > ½ have electrical quadrupolar moments. 4 € Motional Averaging • Molecular motion • Previously, we used averaging to simplify the Hamiltonian Molecular orientation depends on time and Hamiltonian terms can be written as ˆ 0 H int (Θ(t)). These terms were replaced by their time averages: τ ergodicity Hˆ 0 = 1 Hˆ 0 Θ(t) dt Hˆ 0 Hˆ 0 p d int τ ∫ int ( ) int = ∫ int (Θ) Θ € 0 p(Q)=probability Secular isotropic 1 0 Hamiltonian Isotropic materials: Hˆ Hˆ d density for molecule int = N ∫ int (Θ) Θ normalization having orientation Q € • We no longer want to make this€ approximation. Instead, the time variations will be analyzed as perturbations. € 5 Time-averaged Spin Hamiltonian Relative magnitudes 6 Instantaneous Spin Hamiltonian Relative magnitudes 7 Simplifications • In general, the nuclear spin Hamiltonian is quite complicated. • We’ll regularly make use of two simplifications. 1. For terms in the Hamiltonian that are periodic, we change to a rotating frame of reference. ˆ ˆ ˆ Hˆ ! = e−iωtIz Hˆ = e−iωtIz HˆeiωtIz rotating frame laboratory frame Hˆ t Iˆ Iˆ cos t Iˆ sin t Hˆ Iˆ Iˆ ( ) = −ω0 z −ω1 ( x ω − y ω ) eff = −(ω0 −ω) z −ω1 x 2. The secular approximation 8 B0-Electron Interactions When a material is placed in a magnetic field it is magnetized to some degree and this modifies the field… Electrons in an atom circulate about B0, Shielding: generating a magnetic moment opposing the applied magnetic field. • Global effects: magnetic susceptibility s B0 = (1− χ)B0 field inside sample bulk magnetic susceptibility applied field Hereafter well use B0 to refer to the internal field. € • Local effect: Chemical Shift shielding constant (Dont confuse with the Different atoms experience spin density operator!) different electron cloud densities. B = B0(1−σ) 9 € The Zeeman Hamiltonian ! • The interaction energy between the magnetic field, , and the ! ! B magnetic moment, µ = γ I , is given by the Zeeman Hamiltonian. ! ! ! !ˆ Classical: E = −γB ⋅ I QM: Hˆ = −γB ⋅ I zeeman€ • The formal€ correction for chemical shielding is: !ˆ ! Hˆ I (1 )B where σ = 3× 3 shielding tensor € zeeman = −γ −σ € • In vivo, rapid molecular tumbling averages out the non-isotropic components. σ = σ iso = Tr(σ /3) € ! € • Hence for B = [0, 0, B0 ]: E ˆ ˆ €H Zeeman = −γ(1−σ )B0Iz B = 0 B = B0 10 € € A little foreshadowing… Chemical Shielding Tensor • Electron shielding is in general anisotropic, i.e. the degree of shielding depends on the molecular orientation. • The shielding tensor can be written as the sum of three terms: ! $ σ xx σ xy σ xz ! 1 0 0 $ # & # & See Kowalewski, pp # & (1) (2) 105-6 for details. σ = σ yx σ yy σ yz = σ iso # 0 1 0 &+σ +σ # & # 0 0 1 & symmetric and # σ zx σ zy σ zz & " % traceless " % antisymmetric 1 = 3 (σ xx +σ yy +σ zz ) • Both σ(1) and σ(2) are time-varying due to molecular tumbling. • σ(2) gives rise to a relaxation mechanism called chemical shift anisotropy (CSA). (to be discussed later in the course) • σ(1) causes only 2nd order effects and is typically ignored. 11 J-Coupling: Mechanism • At very small distances (comparable to the nuclear radius), the dipolar interaction between an electron and proton is replaced by an isotropic interaction called Fermi contract interaction. A simple model of J-coupling Energy Diagram less stable I e e S I spin senses polarization of more stable S spin I e e S !ˆ !ˆ independent of molecular orientation • Interaction energy ∝−γ γ I ⋅ S e n Scalar coupling • What’s in a name? J-coupling Spin-spin coupling € Through-bond (vs through- space) interaction Indirect interaction 12 A little foreshadowing… J-Coupling and Relaxation • Because J is unchanged with molecular tumbling, J-coupling typically does not contribute to relaxation. ! ! ˆ ˆ ˆ where J Θ t = J H J = 2π JI ⋅ S ( ( )) • However, there are a few cases where J can become “effectively” time-varying. - Case 1: the S spin is engaged in chemical exchange - Case 2: the T1 of the S spin itself is << 1/J. • These cases are called scalar relaxation of the first and second kind respectively, and both are important for the study of MRI contrast agents. 13 Magnetic Dipoles • Nuclei with spin ≠ 0 act like tiny magnetic dipoles. # µ0 &# µ & Bµx = % (% (( 3sinθ cosθ) $ 4π '$ r3 ' permeability of free space Bµy = 0 # µ &# µ & B = % 0 (% ( 3cos2 θ −1 € µz $ 4π '$ r3 '( ) falls off as r3 Dipole at origin € Magnetic Field in y=0 plane € Lines of Force Bµz Bµx 14 Dipolar Coupling • Dipole fields from nearby spins interact (i.e. are coupled). • Rapid fall off with distance causes this to be primarily a intramolecular effect. Spins remain aligned with B0 Water with molecule tumbling in a magnetic field Interaction is time variant! 15 The Nuclear Dipolar Coupling Hamiltonian • Mathematically speaking, the general expression is: µ γ γ # "ˆ !ˆ 3 !ˆ ! !ˆ ! & ! ˆ 0 I S where r vector from Hdipole = − 3 !%I ⋅ S − 2 (I ⋅ r)(S ⋅ r)( 4πr $ r ' spin I to spin S • Secular approximation: $ !ˆ !ˆ ' µ γ γ 2 Hˆ = d& 3Iˆ Sˆ − I ⋅ S ) where d = − €0 I S " 3cos Θ −1 dipole % z z ( 3 ( IS ) 4πr dipole coupling angle between B0 constant and vector from spins I and S ˆ € - With isotropic tumbling, the time average of Hdipole = 0 - However, the temporal variations of ˆ are typically Hdipole (t) the dominant source of T1 and T2 relaxation in vivo. 16 Quadrupolar Interactions • Nuclei with spin I > ½ have a electrical quadrupolar moment due to their non-uniform charge distribution. • This electrical quadrupole moment interacts with local electric field gradients - Static E-field gradients results in shifts of the resonance frequencies of the observed peaks. - Dynamic (time-varying) E-field gradients result in relaxation. • Quadrupolar coupling Hamiltonian (secular approximation): Coupling constant ! ! Looks like an ˆ 3eQ ˆ2 ˆ ˆ HQ = V0 3Iz − I ⋅ I interaction of a spin 4I (2I −1)! ( ) with itself. What’s the spin of Gd3+ with its 7 unpaired electrons? Electric field gradient – dependent on molecular orientation 17 A little foreshadowing… Nucleus-unpaired electron couplings • Both nuclear-electron J and dipolar coupling occur. • Important for understanding MR contrast agents. Magnevist 7 unpaired electrons, I = 7/2, non-zero quadrupolar moment J coupling Dipolar coupling 18 Polarization Single Ensemble of Spins Ensemble of Spins spin Example 1 Example 2 B B “Professor Bloch has told you how one can detect the precession of the magnetic nuclei in a drop of water. Commonplace as such experiments have become in our laboratories, I have not yet lost a feeling of wonder, and of delight, that this delicate motion should reside in all the ordinary things around us, revealing itself only to him who looks for it. I remember, in the winter of our first experiments, just seven years ago, looking on snow with new eyes. There the snow lay around my doorstep - great heaps of protons quietly precessing in the earth’s magnetic field.” - Edward Purcell, Nobel Lecture 1952 • In tissue, we are always dealing with a large number of nuclei, and ! ! the net magnetization is given by: M = ∑µ volume 19 € Phase Coherence Arrows represents spin polarization axis 20 Two-spin Phase Coherences Sˆ Iˆ z I x S € Net tendency for S spins to be +z€ No net tendency for S spins in any direction No net tendency for I spins in any direction Net tendency for I spins to be +x 2Iˆ Sˆ z z ? ? € No net tendency for I or S spins to be z If I or S is z, increased probability paired spin is z 21 Magnetization = Phase Coherences • Some coherences are …while others are not.